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A Hybrid Stochastic Deterministic Algorithm for Solving Unconstrained Optimization Problems

Author

Listed:
  • Ahmad M. Alshamrani

    (Statistics and Operations Research Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Adel Fahad Alrasheedi

    (Statistics and Operations Research Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Khalid Abdulaziz Alnowibet

    (Statistics and Operations Research Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

  • Salem Mahdi

    (Department of Mathematics & Computer Science, Faculty of Science, Alexandria University, Alexandria 21545, Egypt
    Educational Research and Development Center Sanaa, Sanaa 00967, Yemen)

  • Ali Wagdy Mohamed

    (Operations Research Department, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt)

Abstract

In this paper, a new deterministic method is proposed. This method depends on presenting (suggesting) some modifications to existing parameters of some conjugate gradient methods. The parameters of our suggested method contain a mix of deterministic and stochastic parameters. The proposed method is added to a line search algorithm to make it a globally convergent method. The convergence analysis of the method is established. The gradient vector is estimated by a finite difference approximation approach, and a new step-size h of this approach is generated randomly. In addition, a set of stochastic parameter formulas is constructed from which some solutions are generated randomly for an unconstrained problem. This stochastic technique is hybridized with the new deterministic method to obtain a new hybrid algorithm that finds an approximate solution for the global minimization problem. The performance of the suggested hybrid algorithm is tested in two sets of benchmark optimization test problems containing convex and non-convex functions. Comprehensive comparisons versus four other hybrid algorithms are listed in this study. The performance profiles are utilized to evaluate and compare the performance of the five hybrid algorithms. The numerical results show that our proposed hybrid algorithm is promising and competitive for finding the global optimum point. The comparison results between the performance of our suggested hybrid algorithm and the other four hybrid algorithms indicate that the proposed algorithm is competitive with, and in all cases superior to, the four algorithms in terms of the efficiency, reliability, and effectiveness for finding the global minimizers of non-convex functions.

Suggested Citation

  • Ahmad M. Alshamrani & Adel Fahad Alrasheedi & Khalid Abdulaziz Alnowibet & Salem Mahdi & Ali Wagdy Mohamed, 2022. "A Hybrid Stochastic Deterministic Algorithm for Solving Unconstrained Optimization Problems," Mathematics, MDPI, vol. 10(17), pages 1-26, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:17:p:3032-:d:895159
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    References listed on IDEAS

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    1. Gonglin Yuan & Zehong Meng & Yong Li, 2016. "A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 129-152, January.
    2. Khalid Abdulaziz Alnowibet & Salem Mahdi & Mahmoud El-Alem & Mohamed Abdelawwad & Ali Wagdy Mohamed, 2022. "Guided Hybrid Modified Simulated Annealing Algorithm for Solving Constrained Global Optimization Problems," Mathematics, MDPI, vol. 10(8), pages 1-25, April.
    3. Y.H. Dai & Y. Yuan, 2001. "An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimization," Annals of Operations Research, Springer, vol. 103(1), pages 33-47, March.
    4. Zhenhua Su & Min Li, 2020. "A Derivative-Free Liu–Storey Method for Solving Large-Scale Nonlinear Systems of Equations," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-10, October.
    5. Yingjie Zhou & Yulun Wu & Xiangrong Li, 2020. "A New Hybrid PRPFR Conjugate Gradient Method for Solving Nonlinear Monotone Equations and Image Restoration Problems," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-13, September.
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